Skip to content
Exit presentation

I'm Up and Down, and Round and Round

Track quiz

Thirteen ideas, thirteen checks. The table comes first.

Thirteen ideas in short
IdeaHow it goes
The four namesChord straight, arc curved, segment one chord, sector two radii.
Chord and centreTake a chord that is not a diameter. A 90° line from the centre meets it at its middle.
Length and distanceBigger distance out, smaller half-length. Further out is shorter.
TwiceThe centre angle is double the edge angle on the same arc.
Same arcEvery point on one arc gives the same angle.
On a diameter180° at the centre gives exactly 90° at the rim.
Cyclic quadrilateralOpposite angles total 180°.
Concyclic testFix the circle with three points. Measure the fourth.
CircumcentreTwo bisectors cross there. Acute inside, right on the side, obtuse outside.
Two pointsA circle passes through A and B. Its centre lies on the perpendicular bisector of AB.
TurningA circle looks the same after any turn about its centre.
Equal chordsIn one circle, equal angles at the centre give equal chords.
One sideAB makes equal angles at C and D. C and D are on the same side of line AB. Then A, B, C and D lie on one circle.

Which is a piece of the circle itself? Not a region or a line.

  • An arc
  • A sector
  • A chord

A circle has radius 26 cm and a chord 20 cm long. How far is the chord from the centre?

  • 16 cm, from 26 take away 10
  • 24 cm, since 676 − 100 = 576
  • 10 cm, which is half the chord

Two chords sit 5 cm and 12 cm from a circle's centre. Which is longer?

  • The one 12 cm out, since it reaches further
  • They are equal, since the circle is the same
  • The one 5 cm out, since less is taken from the radius

An arc makes 150° at the centre. What does it make at the rim?

  • 300°, since the rim angle is the wider one
  • 75°, since 150 ÷ 2 = 75
  • 150°, since it is the same arc

P and Q both sit on chord AB's major arc. Compare angle APB with angle AQB.

  • They are equal, since both halve the same centre angle
  • Angle APB is bigger, since P is nearer the chord
  • Angle AQB is bigger, since Q sees more of the circle

A triangle has all three corners on a circle. One side is a diameter. What is the angle opposite that side?

  • About 90°, depending on the corner
  • Exactly 90°, since 180 ÷ 2 = 90
  • 180°, since a diameter is a straight line

A four-sided shape has all four corners on a circle. Two corners next to each other read 95° and 105°. What is the angle opposite the 95° one?

  • 265°, since the two opposite ones must total 360°
  • 105°, since opposite angles match
  • 85°, since 180 − 95 = 85

Three points fix a circle with centre (2, 1.5) and radius 2.5. A fourth point is 2.5 from that centre. Are the four concyclic?

  • Yes, since the fourth is exactly 2.5 away
  • No, since four points never all fit on one circle
  • There is no way to tell without drawing it

A triangle has angles of 20°, 25° and 135°. Where is its circumcentre?

  • Outside the triangle, since 135° is past 90°
  • Inside the triangle, since the triangle sits inside its circle
  • On the middle of its shortest side

How many circles pass through two given points A and B?

  • Exactly one, since two points fix the centre and the radius
  • Exactly two, since the centre can be on either side of AB
  • Endless, since each point on the bisector can be a centre

A regular pentagon is turned about its centre. What is the smallest turn that fits it onto itself?

  • 60°, since 360 ÷ 6 = 60
  • 72°, since 360 ÷ 5 = 72
  • 90°, since a square fits after 90°

AB and DE are chords of one circle. Each makes 80° at the centre. AB is 12 cm. How long is DE?

  • Not known, since the radius is not given
  • Shorter, since DE sits lower down in the circle
  • 12 cm, since equal angles give equal chords

AB makes 48° at C and 48° at D. C and D are on the same side of line AB. What follows?

  • Nothing follows, since four points need not share a circle
  • All four lie on one circle, since the angles are equal
  • Only A, B and C share a circle, since D is new

This chapter is Chapter 5. It is called I'm Up and Down, and Round and Round.

Next: Measuring Space: Perimeter and Area. The way round a circle, and the space inside it.

Jump to lesson

This track

  1. 0 Start here
  2. 1 Four names for four parts of a circle
  3. 2 The line from the centre that cuts a chord in half
  4. 3 Same distance, same length — and further out is shorter
  5. 4 An arc makes twice the angle at the centre that it makes at the edge
  6. 5 Every point on one arc sees the chord the same way
  7. 6 Checkpoint: parts, chords and the doubling
  8. 7 A corner standing on a diameter is a right angle
  9. 8 Four corners on one circle, and the pairs that total 180°
  10. 9 Testing whether four points really share one circle
  11. 10 Three points give one circle, and the triangle shows where its centre is
  12. 11 Checkpoint: diameters, cyclic shapes and finding a centre
  13. 12 Many circles fit through two points
  14. 13 A circle looks the same when turned
  15. 14 Equal angles at the centre give equal chords
  16. 15 Equal angles on one side put four points on a circle
  17. 16 The big picture
  18. 17 Track quiz

Other tracks